Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,6)(3,4,7,8)(11,15)(14,16)(19,23)(22,24);(2,6)(3,7)(10,13)(11,15)(17,18,21,20)(19,22,23,24);(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24);(1,3)(2,4)(5,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,20)(19,22,23,24);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,24,23,22)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #130188, #485768, #485840, … |
| maximal normal subgroup of | #130188, #485768, #485840, … |
| maximal quotient of | #557165, #557483, #557502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38732);